Trotsky was not some lunatic adventurer - he wouldn't invade Eastern Europe, as he knows that would provoke a united pan-European anti-communist crusade against the USSR.
I've stated before, while the theory of Permanent Revolution is opposed to Socialism in One Country, it doesn't mean that Trotsky was dumb enough to invade a swathe of neighbouring countries, which he knows is likely to provoke a grand coalition against Bolshevism. He knows the Soviet Union is a pariah. In fact, during the interwar period Trotsky advocated that the Comintern should be encouraging foreign Communist parties to adopt United Front tactics (not Stalin's bullshit "Anti-Social Fascism Position"), that is to ally with other leftist/progressive movements (e.g. Social Democrats) in their home countries to oppose fascism (not communism being imposed at the tip of Soviet bayonets with a foreign invasion). Trotsky also denounced Stalinist autarky, and instead encouraged trade and technological exchanges with the advanced Western capitalist countries.
While Stalin (and to a lesser extent Zinoviev) are alive, it is extremely difficult for Trotsky to take power, as both of them were significant obstacles and much better political schemers (moreso the former). The best way for Trotsky to take power is for Stalin and Zinoviev to die during the Russian Civil War, and then for Lenin to live longer (possibly due to avoiding getting hit by Fanny Kaplan's bullets). Yes Trotsky was extremely arrogant and had a knack for alienating peers because of it, but he was also basically the co-leader of the Bolshevik coup and all the other contenders - Stalin, Zinoviev and Bukharin - identified him as the main threat and banded together against him. It took their combined efforts and Stalin's unparalleled Machiavellianism to take him down. So without Stalin and Zinoviev as key obstacles against him, and with the backing of Lenin, I think Trotsky has a very good shot at leadership.
However, I don't see Trotsky ever wielding the kind of absolute power like Stalin had IOTL. Soviet leadership under Trotsky would be far more collegiate and collectivist - rule by committees and troikas - rather than a single totalitarian absolute dictator like under Stalinism, simply because Trotsky has neither the inclination nor the sheer political skill to become a totalitarian dictator.
With the lack of Stalin and Zinoviev, opposition to Lenin and Trotsky in the Party would coalesce around the Worker's Opposition of Kollontai and the Right Opposition of Bukharin, Rykov, and Tomsky (and both factions have far less clout and political ability than Stalin). Also, a longer-lived Lenin would be able to thoroughly pave the way for Trotsky to be his official successor, but with checks and balances.
So lets say Stalin and Zinoviev die in the RCW, and Fanny Kaplan's bullets miss Lenin during her assassination attempt, so Lenin lives a few years longer and dies in the 1930s after officially anointing Trotsky his successor. But the Soviet Union comes under a committee-style of leadership where Trotsky is only first among equals
Here are the main differences between OTL's Stalinism and Trotsky as leader of the Soviet Union:
- No Stalinist Great Purge or Moscow Show Trials. Some Bolsheviks who still disagree too vehemently with the Party-line after it has been agreed via Democratic Centralism would be forced into retirement (or even imprisoned or exiled), but the ban on factions would be lifted and there would no mass killings within the Party, Soviet government or Red Army.
- However, the Red Terror would continue against "counterrevolutionary reactionaries" outside the Party.
- Most of the gulag camps and slave labor system would be gradually phased out, but a few would continue to exist for "enemies of the people" and some forced labour would be utilised. However, mass deportations (especially of entire ethnic groups) would not occur.
- A much more restrained Cult of Personality than Stalin's IOTL one probably occurs, focused on the duarchy of Lenin-Trotsky and the latter as the former's heir. Marxist-Leninism (ironically also known as Trotskyism ITTL) is probably codified.
- Centrally planned industrialization and agricultural collectivization would be implemented, but not with Stalin's sheer level of incompetence/bungling, zig-zagging, unpreparedness and targeted racism against Ukrainians, so millions won't die in a Holodomor and brutal slave labor. Industrialisation is also more effective with Trotsky encouraging trade and technology/skill swaps with the Western world, rather than Stalinist autarky.
- The Red Army would be less professionalized than under Stalinism, and would instead be more like the "people's militia model" advocated by Trotsky IOTL. The Red Army may be weaker in some respects, but would still receive heavy investment and be vastly better off overall without the Great Purge and Stalin's bungling (Trotsky was the father of the Red Army and a skilled military man - the "Red Napoleon").
- No Socialism in One Country. The Soviet Union would continue to be ideologically committed to internationalist proletarianism, global working-class solidarity, and worldwide revolution. Trotsky would tone down advocating permanent revolution in Europe though, as he knows that will provoke an anti-communist crusade against the Soviet Union and the failed revolutions in Germany, Hungary, etc, proves that Europe "isn't ready". However, the Comintern would not be purged, neutered and abolished like happened IOTL under Stalinism; instead foreign communist parties would receive strong support, but more scope for independent action, from Moscow. The Internationale would remain the Soviet Union's national anthem.
- Trotsky would encourage European communist parties to form anti-fascist Popular/United Fronts with progressive, anarchist, and social democratic parties and cooperate with them against fascism (i.e. no Stalinist haranguing of "social fascism"). The Spanish Republicans and the KPD will be much better off for it.
- Trotsky would neglect communist/anti-colonial movements in the Third World, and will focus on the urban proletariat instead of the much more numerous peasantry. This isn't much different from interwar Soviet policy in the Third World from IOTL though.
- However, Trotsky would probably encourage the Chinese Communists to disengage from the united front with the Kuomintang and instead maintain their independence, including retaining an independent armed militia. This may save the Chinese Communists from Chiang Kai-Shek's 1927 Shanghai purge. But, Trotsky would not approve of Maoist theories on peasant revolution.
- No Russification. Ethnic minorities and the SSRs within the Soviet Union would have more autonomy than under Stalinism, and would not have Russian culture/language forced on them. No ethnically targeted, racist annihilation campaigns like Stalin's Polish Operation, Korean Operation, mass deportations of minorities, etc.
- State Atheism and anti-clerical persecution would continue, and there would be no de-facto alliance with the Russian Orthodox Church like under Stalinism.
- The Party would try and stamp out anti-Semitism instead of encouraging it like Stalin did IOTL, so no Doctor's Plot/"rootless cosmopolitanism", and skilled Jewish immigration to the Soviet Union would be encouraged. Many socialist-leaning Jews may end up fleeing to the friendly Soviet Union as refugees from fascism.
- There is no way in hell that Trotsky is ever trusting of pacts with fascist powers like Nazi Germany, especially intensely anti-Semitic dictators like Hitler (who would hate the Jewish Trotsky with a passion). So the Molotov-Ribbentrop Pact never happens. Trotsky wouldn't be as gullible as Stalin either - he would expect the fascists to attack the Soviet Union, and prepare accordingly.